Let $G$ be an algebraic group acting on a variety $V$. Which information can be obtained by looking the action of $G$, and subgroups of $G$ that fixes points of $V$?. In other words how we obtain $V$ from the group $G$?

3 Answers
3

A lot if $G$ is transitive. Then $V=G/H$ for a subgroup $H$ (if it has a point), or a $G$-torsor mod $H$ (if it doesn't). Then most questions about the geometry of the variety are best answered by studying the group action. For instance, we can study line bundles on a flag variety of a reductive group using the root lattice for that group.

Another case where you gain a lot of information is where $G$ acts almost transitively, i.e., there is a dense orbit, as in the case of toric varieties. Then it is not as simple to "obtain" our variety as just choosing a group $G$ and subgroup $H$. We must also include some information on how to glue on the other orbits. But usually, because of the extra symmetry the group structure provides, this description is not so complex as defining an entire algebraic variety, since if you know something about the geometry of a point, you can deduce the equivalent statement about all other points in its orbit.

What about if stabilizer of a point is solvable?. Can we know someting about this variety?. For example its cohomology groups?
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albert cohenDec 22 '12 at 18:18

I don't know any theorems with only that condition. What do you know about the group?
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Will SawinDec 22 '12 at 19:42

1

@Will: It seems that you mean that if an algebraic group $G$ defined over a field $k$ acts transitively on a $k$-variety $V$, then $V$ can be dominated by a principal homogeneous space (torsor) $P$ of $G$. However, this is not true in general. The action of $G$ on $V$ defines a certain second nonabelian cohomology class $\eta(V)$ of ${\rm Gal}(\bar k/k)$ with coefficients in a band (lien) related to the stabilizer $\overline{H}$ of a $\bar k$-point of $V$. Such a torsor $P$ exists if and only if this cohomology class is neutral.
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Mikhail BorovoiJan 1 '13 at 16:27

As indicated by the other answers, you question is not specific enough.

In the particular case where $G$ is the multiplicative group, a theorem of Białynicki-Birula (On fixed point schemes of actions of multiplicative and additive groups.
Topology12 (1973), 99–103, MR:313261), furnishes a decomposition of $V$ into locally closed subsets $V_i$, each of them being stable under the action of $G$ and a trivial fibration over the fixed point set $V_i^G$.